2.11. The Time Inversion Operation θ

The time inversion operation θ [17]
(pp. 325-348) is of interest here principally in connection with intensity
calculations. The intensity of absorption or emission of light by diatomic
molecules depends on matrix elements of the components of the dipole moment
operator. These matrix elements are usually not calculated explicitly, but are
rather treated as parameters, to be determined from a fit to experimental data.
It sometimes happens that several such parameters occur in the intensity
expressions, which then involve, for example, the squares of sums and
differences of these parameters. It is clearly desirable to know which of the
parameters are real and which are complex, since the arithmetic of real numbers
is not identical to the arithmetic of complex numbers. Time inversion is a
useful tool, since it is essentially the operation of taking complex
conjugates. Indeed,

(2.27)

if k is a constant or a function of the positional coordinates of
particles. However, because of the rather special nature of spin variables,
time inversion, when applied to spin functions, is somewhat more complicated
[17] (pp. 331-333).

Physically, time inversion would be expected to correspond to a transformation
of variables in which the time t is replaced by - t. Thus,
for example, a position coordinate x should remain invariant under time
inversion, while a velocity dx/dt or a momentum
m(dx/dt) should transform into its negative. In quantum
mechanics momenta are represented by operators of the form
- i(d/dx),
which do not contain the time variable at all. However, in contrast to position
coordinates, they do contain the pure imaginary number i. It is thus
convenient in quantum mechanics to construct a formalism in which time
inversion corresponds to the taking of complex conjugates rather than the
replacing of t by - t.

Arguments such as this make the following transformation equations for angular
momentum operators seem reasonable. (They are also correct
[17] (pp. 329-330).)

When the system being considered contains an even number of electrons,
θ2 = +1 [17] (p. 332). Under these
circumstances, it happens that L, S, and J are all whole
numbers, so that zero is a possible value for each of the projection quantum
numbers Λ, Σ, Ω, and M. It is relatively easy to show
that the phase factor of the wave function having a projection quantum number
equal to zero can be chosen such that the function is unchanged when the time
inversion operation is carried out. In other words, it is always possible to
choose phases such that

When the system being considered contains an odd number of electrons,
θ2 = -1 [17]
(p. 332). Under these circumstances. S and J are
half-integers, so that a value of zero for Σ, Ω, and M is
not possible. There are then no spin functions and no rotational functions
which remain unchanged by the time inversion operation [see
(2.30)]. However, it is possible to show that one can
always choose phases consistent with those of Condon and Shortley
[7], such that

A set of four equations similar to (2.34) holds for the functions when
J is half-integral, except that both laboratory-fixed
(JX ± iJY) and
molecule-fixed (JxiJy) ladder operators must be used.
One of these equations takes the form

where m and n are positive integers and k3 is a
positive constant. This equation and the three analogous equations obtained by
using different combinations of (JxiJy)m and
(JX ±
iJY)n can be shown to be equivalent to